Homological stability for spaces of embedded surfaces

Abstract

We study the space of oriented genus-$g$ subsurfaces of a fixed manifold $M$ and, in particular, its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over $M$ associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees.

Our results are analogous to McDuff’s theorem on configuration spaces, extended from $0$–dimensional submanifolds to $2$–dimensional submanifolds.